Elysium's improvements to the world would be less about the use of their resources and more about sharing technology and know how.
They understand efficient use of closed ecosystems.
All real world attempts to create a self sustaining closed ecosystem haven't lasted very long. Even if Elysium isn't 100% closed, the fact is if they don't need regular food shipments, they're doing something great, and that know-how could drastically improve the life of people on Earth, if used for crops, medicine, and livestock.
When your robots are no longer being used as violent overseers, but actually to do labor, well, that's a mass of labor output you've gained. Better housing, structures, can be made with the materials available.
The populace can instead be focused on education and directed towards research in further advances in ecological science, biotech, science, etc. to continue the tech advancements necessary to restore Earth. The reeducation might not be as difficult as one might suspect - after all, it's clear in the movie that the gang members had people on their team who could handle spaceflight, cryptography, and more...
Potential decontamination projects
If their medicine can pull out toxins or contaminants from a human body, who's to say it can't be pulled out of water, soil or plants? That might be a long term project given the scope of pollution shown in the movie, but it'd be better than nothing.
If you combine this technology with various plant species which are known to suck up heavy metals, radioactive materials and other toxic substances, cleanup becomes much easier.
Easy Space Travel means Easy Disposal of Waste
If trips up to space are short, easy, and regular, there's no reason to leave the unfixable toxic materials on Earth. We can simply launch them away from the planet, or, if for some reason we might want to recycle them later, put a storage base on the moon.
The biggest problem is that effectively Earth has been a repressed dictatorship for quite some time. When you rule a country with repressive force, and remove that, it's very hard to stabilize right after - there's a lot of old bad blood between everyone, and a tendency to replicate the violence everyone's lived under for so long.
So the first issue is not having things devolve into a bloodbath. Assuming you manage that, and get to a stable government (whether that's a single world government, or several countries again), then you have to figure out how to institute the changes. Some people will be suspicious of the new set up, some will try to take power and keep others out, etc.
The problems are political because the root issue was political: any group that had enough power to forcibly repress the entire population with robots COULD have instead used that power to try to stop the pollution, set up sensible policies both in birth control, education and food access instead.
Any research done on Elysium about "fixing the Earth" are probably also mired into genocidal eugenics type plans as well - ethically unfeasible and probably politically explosive to even hear about for the public.